Worksheet: Straight Line Equations in Vector Form

In this worksheet, we will practice finding the equation of a straight line in vector form.

Q1:

Which of the following can be the vector form of the equation of the straight line 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 = 0 , where 𝑎 0 and 𝑏 0 ?

  • A 𝑟 = 0 , 𝑐 𝑏 + 𝐾 ( 𝑏 , 𝑎 )
  • B 𝑟 = 𝑐 𝑎 , 0 + 𝐾 ( 𝑎 , 𝑏 )
  • C 𝑟 = 𝑐 𝑎 , 0 + 𝐾 ( 𝑎 , 𝑏 )
  • D 𝑟 = 0 , 𝑐 𝑏 + 𝐾 ( 𝑏 , 𝑎 )
  • E 𝑟 = 𝑐 𝑎 , 0 + 𝐾 ( 𝑎 , 𝑏 )

Q2:

Which of the following can be the vector form of the equation of the straight line 𝑥 𝑎 + 𝑦 𝑏 = 1 , where 𝑎 0 and 𝑏 0 ?

  • A r = 1 𝑎 , 0 + 𝐾 𝑏 , 𝑎
  • B r = 0 , 𝑏 + 𝐾 𝑎 , 𝑏
  • C r = 0 , 1 𝑏 + 𝐾 𝑏 , 𝑎
  • D r = 𝑎 , 0 + 𝐾 𝑎 , 𝑏
  • E r = 0 , 1 𝑎 + 𝐾 ( 𝑏 , 𝑎 )

Q3:

Find the vector equation of the straight line passing through the origin and the point ( 0 , 4 ) .

  • A r = 0 , 4
  • B r K = 4 , 0
  • C r = 4 , 0
  • D r K = 0 , 4

Q4:

Find the vector equation of the straight line passing through the points ( 6 , 7 ) and ( 4 , 6 ) .

  • A r = 6 , 4 + 𝐾 7 , 6
  • B r = 4 , 6 + 𝐾 1 3 , 1 0
  • C r = 4 , 6 + 𝐾 1 0 , 1 3
  • D r = 6 , 7 + 𝐾 1 0 , 1 3

Q5:

Find the vector equation of the straight line whose slope is 8 3 and passes through the point ( 4 , 9 ) .

  • A r = 4 , 9 + 𝐾 8 , 3
  • B r = 9 , 4 + 𝐾 3 , 8
  • C r = 3 , 8 + 𝐾 4 , 9
  • D r = 4 , 9 + 𝐾 3 , 8

Q6:

Consider the line through the point ( 3 , 6 ) and perpendicular to vector r = 3 , 6 . Which of the following is a vector equation of this line?

  • A 𝐾 = 3 , 6 + 6 , 3 r
  • B r = 3 , 6 + 𝐾 3 , 6
  • C 𝐾 = 3 , 6 + 3 , 6 r
  • D r = 3 , 6 + 𝐾 6 , 3
  • E r = 6 , 3 + 𝐾 3 , 6

Q7:

Find the vector form of the equation of the straight line passing throught the point 𝐴 ( 2 , 5 , 5 ) and parallel to the straight line passing through the two points 𝐵 ( 3 , 2 , 6 ) and 𝐶 ( 5 , 0 , 9 ) .

  • A r = 2 , 5 , 5 + 𝑡 3 , 2 , 6
  • B r = 8 , 2 , 3 + 𝑡 2 , 5 , 5
  • C r = 2 , 5 , 5 + 𝑡 5 , 0 , 9
  • D r = 2 , 5 , 5 + 𝑡 8 , 2 , 3

Q8:

Write the vector equation of the straight line that passes through the point 6 , 9 with direction vector 9 , 2 .

  • A K r = 6 , 9 + 9 , 2
  • B r K = 9 , 2 + 6 , 9
  • C K r = 9 , 2 + 6 , 9
  • D r K = 6 , 9 + 9 , 2

Q9:

Find the vector form of the equation of the straight line passing through the point ( 5 , 1 , 4 ) and the intersection point of the two straight lines 𝑥 + 2 2 = 𝑦 + 5 2 = 𝑧 + 3 1 and 𝑥 + 1 3 = 𝑦 1 2 = 𝑧 + 3 2 .

  • A r = 5 , 1 , 4 + 𝑡 7 , 2 , 9
  • B r = 5 , 1 , 4 + 𝑡 7 , 2 , 9
  • C r = 5 , 1 , 4 + 𝑡 7 , 2 , 9
  • D r = 5 , 1 , 4 + 𝑡 7 , 2 , 9

Q10:

Find the vector form of the equation of the straight line passing through the point ( 2 , 5 , 5 ) and the centre of the sphere whose equation is 2 𝑥 + 2 𝑦 + 2 𝑧 + 1 2 𝑥 8 𝑦 + 8 𝑧 = 1 .

  • A r = 2 5 5 + 𝑡 5 7 3
  • B r = 2 5 5 + 𝑡 1 2 8 8
  • C r = 2 5 5 + 𝑡 1 2 8 8
  • D r = 2 5 5 + 𝑡 5 7 3

Q11:

Determine the vector form of the equation of the straight line passing through the point ( 1 , 5 , 4 ) and parallel to the vector 3 , 5 , 1 .

  • A r = 1 , 5 , 4 + 𝑡 2 , 1 0 , 3
  • B r = 3 , 5 , 1 + 𝑡 1 , 5 , 4
  • C r = 3 , 5 , 1 + 𝑡 2 , 1 0 , 3
  • D r = 1 , 5 , 4 + 𝑡 3 , 5 , 1

Q12:

Determine, in vector form, the equation of the straight line that passes through the points ( 5 , 5 , 3 ) and ( 3 , 4 , 4 ) .

  • A r = 2 , 1 , 1 + 𝑡 5 , 5 , 3
  • B r = 5 , 5 , 3 + 𝑡 8 , 9 , 7
  • C r = 2 , 1 , 1 + 𝑡 3 , 4 , 4
  • D r = 5 , 5 , 3 + 𝑡 2 , 1 , 1

Q13:

Find the vector equation of the straight line that is parallel to the 𝑥 -axis and passing through the point ( 5 , 2 ) .

  • A r K = 5 , 2 + 0 , 1
  • B r K = 2 , 5 + 1 , 0
  • C r K = 2 , 5 + 0 , 1
  • D r K = 5 , 2 + 1 , 0

Q14:

Consider the line through the point ( 0 , 4 ) and perpendicular to vector r = 0 , 4 . Which of the following is a vector equation of this line?

  • A 𝐾 = 0 , 4 + 4 , 0 r
  • B r = 0 , 4 + 𝐾 0 , 4
  • C 𝐾 = 0 , 4 + 0 , 4 r
  • D r = 0 , 4 + 𝐾 4 , 0
  • E r = 4 , 0 + 𝐾 0 , 4

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